Maximizing Singlet Fission in Organic Dimers: Theoretical Investigation of Triplet Yield inthe Regime of Localized Excitation and Fast Coherent Electron Transfer†

Eric C. Greyson,‡ Josh Vura-Weis,‡ Josef Michl,*,§,| and Mark A. Ratner‡

Department of Chemistry, Northwestern UniVersity, 2145 Sheridan Road, EVanston, Illinois 60208-3113,Department of Chemistry and Biochemistry, UniVersity of Colorado, 215 UCB, Boulder, Colorado 80309-0215,and Institute of Organic Chemistry and Biochemistry, Academy of Sciences of the Czech Republic,FlemingoVo n. 2, 166 10 Prague 6, Czech Republic

ReceiVed: July 31, 2009; ReVised Manuscript ReceiVed: December 20, 2009

In traditional solar cells one photon absorbed can lead to at most one electron of current. Singlet fission, aprocess in which one singlet exciton is converted to two triplet excitons, provides a potential improvementby producing two electrons from each photon of sufficient energy. The literature contains several reports ofsinglet fission in various systems, but the mechanism of this process is poorly understood. In this paper weexamine a two-step mechanism with a charge transfer state intermediate, applicable when the initial excitedstate is localized. Density matrix theory is used to examine how various molecular properties such as orbitalenergies and electronic couplings affect singlet fission yield in the regime of fast, coherent electron transfer.Several promising chromophores are discussed and density functional theory is used to predict fission yieldfor each in the context of this mechanism. Finally, implications for chromophore design are discussed, andfuture experiments are suggested.

Introduction

Singlet fission (SF) was first discovered during an investiga-tion of fluorescence quenching in anthracene crystals forty yearsago.1 Evidence that singlet excitation split into two tripletexcitations was found in the specific magnetic field dependenceof SF efficiency.2-8 A number of more recent studies haveidentified singlet fission in other crystalline systems,9-18 and inseveral polymers and small molecules,19-33 although the fissionmechanism may be difficult to identify conclusively.34

Photovoltaics could benefit from SF by yielding two chargecarriers from a single absorbed photon,35 and the phenomenonhas recently attracted renewed interest. One of the first steps36

was the identification of chromophores whose excitation energyinto the lowest triplet level T1 was roughly half the excitationenergy into the first excited singlet energy level S1, making SFthermodynamically favorable; additionally, the next triplet levelT2 was above S1, making triplet-triplet annihilation endoergic.The energy levels of the individual chromophores are not theonly factor governing SF yield, however, as several papers haveshown very different yields for similar chromophores thatare connected and electronically coupled in differentways.21,25,26,29,30,37,38 In separate work, we used quantum chem-istry to examine how coupling two isolated chromophoresthrough a covalent bond to form a coupled chromophore pairwould affect the free energy of SF and the electronic couplingmatrix element.39 One of the most important conclusions of thiswork is that stronger electronic coupling of the chromophorescan lead to a pair in which SF is less exothermic, or moreendothermic. Because electronic events are most likely to occurif they have large electronic matrix elements and are energeti-cally favorable, there is probably a specific level of coupling

that will optimize SF for any particular chromophore pair. Whilethe understanding of SF is steadily growing, it is not yet clearexactly what balance of molecular properties will result in adimer with the most efficient SF. The present paper deals withdimeric chromophores in which the lowest singlet excitation islocalized and resides on one of the chromophores. As such, itis more applicable to chromophore pairs in solution or in glassthan in crystals.

We examine a three-step mechanism consisting of an initiallocalized absorption on one chromophore followed by electrontransfer from the excited chromophore to its partner, forming acharge transfer intermediate, followed by transfer of an electronof opposite spin in the opposite direction, resulting in a pair ofcharge-neutral triplets (if each chromophore is consideredindependent in the final state). We neglect the direct couplingof the initial state to the final state by the two-electron part ofthe Hamiltonian. Such Coulomb and exchange interaction isbelieved to be important in molecular crystals.40-42 We analyzeevery possible electronic configuration in a four electron, fourorbital model of a coupled chromophore pair and reduce thecomplexity of the problem to the most important configurations.The dynamics of the excitation are modeled with density matrixtheory, assuming fast coherent electron transfer.

To predict SF yield in a series of dimeric chromophores, we(i) created a ten configuration dynamic model by analyzing SFwith a four electron, four orbital basis, (ii) used electronicstructure computations to evaluate both the energy of every stateand electronic coupling element for a few promising chro-mophores, and (iii) applied the density matrix formalism toexamine the evolution of the excitation over time for severaltest cases as well as for real molecules. Each of these threesteps is discussed in further detail below.

Methods

Model System. We use the limit of two uncoupled chro-mophores to model SF in molecular coupled pairs We assume

† Part of the “Michael R. Wasielewski Festschrift”.‡ Northwestern University.§ University of Colorado.| Academy of Sciences of the Czech Republic.

J. Phys. Chem. B 2010, 114, 14168–1417714168

10.1021/jp907392q 2010 American Chemical SocietyPublished on Web 02/25/2010

that the initial optical excitation is localized on one chro-mophore. To consider every possible mechanism for SF, weexamine the 70 possible electron configurations that arise fromfour electrons in the four frontier molecular orbitals (HOMOsand LUMOs) of the uncoupled chromophore pair (Figure 1).

When both chromophores are neutral, each could be in oneof six configurations, and so there are thirty six possible electronconfigurations representing a pair of neutral chromophores(labeled 1-36 above.) If a chromophore is a radical cation oranion (1 or 3 electrons in four frontier orbitals), there are onlyfour possible electron configurations for each half. This providesan additional thirty two configurations (37-68 above), sixteenfrom a A+ ·B- · combination and sixteen from the A- ·B+ · .Chromophores that are doubly charged only have one possibleconfiguration, so our dimeric system has two double chargetransfer configurations (69-70 above.)

We assume that only configurations close in energy to theinitial configurations are likely to be accessed by a significantpopulation during the lifetime of the excitation because of theenergy gap law.43 For singlet fission to be efficient it must beexothermic, and there are rare cases where it is stronglyexothermic, so we assume the energy of one singlet should beabout equal to twice the energy of a triplet. For solar cellapplications, the singlet energy should be around 2.2 eV tooptimize the photocurrent. Configurations such as 15, whichcontains two singlet excitations, or 19, with one singlet and onetriplet excitation, can be easily eliminated from consideration,as we would estimate these to have energies of ∼4.4 or ∼3.3eV (note that the terms “singlet” and “triplet” are not actuallycorrect for some of the configurations in Figure 1, which arenot eigenstates of the total spin operator, but we use these labelsand figures for convenience).

Using this logic, we can reduce our system from seventyconfigurations to sixteen. We keep four configurations that areof S1S0 character (17, 23, 27, 28), four that have TT character

(1, 2, 7, 8), and eight single charge transfer (CT) configurationsthat are in their ground electronic state (37, 38, 45, 46, 53, 54,61, 62).

In organics containing elements with atomic numbers lessthan ten, spin-orbit mixing is weak. Spin dipole-dipolecoupling, which also has the ability to mix pure spin states andwhich is important for the understanding of magnetic fieldeffects on triplet fusion and singlet fission in molecular crystals,will here also be neglected.44 In addition, spin-flips in the CTstate are expected to be slow.45,46 Therefore, in discussingexciton fission we ignore configurations with overall spinprojection * 0 that can only be reached by spin-orbit coupling(1, 8, 37, 46, 53, 62). This yields a reduced ten-configurationscheme, as depicted in Figure 2, with configurations renumberedfor convenience. Eight different one electron transfers can bringan excitation from one of the four S1 configurations to one ofthe four CT configurations. Four of these steps involve oneelectron moving from the HOMO of one chromophore to theHOMO of the other, while the remaining four transitions in-

Figure 1. The 70 electron configurations in a 4-electron 4-orbital basis provided by the HOMO and the LUMO on each of a pair of chromophores.The lowest energy configurations are highlighted.

Figure 2. Allowed electron transfers in a 10-configuration system forSF, with definitive configuration numbering.

Singlet Fission in Organic Dimers J. Phys. Chem. B, Vol. 114, No. 45, 2010 14169

volve the transfer of an electron from the LUMO of onechromophore to the LUMO of the other. Such electron transferscan be referred to as “horizontal”. The two TT configurationscan be reached from the four CT configurations through “non-horizontal” electron transfers, by either sending an electron fromthe LUMO of one chromophore to the HOMO of the otherchromophore or vice versa.

In accordance with Koopmans’ theorem, we make theapproximation that orbital character is not affected by orbitaloccupancy. This means that every electron transfer from theHOMO of one chromophore to the HOMO of the otherchromophore (or LUMO to LUMO) has the same matrixelement, TH (TL). For an electron to move from a CT to a TTconfiguration there are four possible electron transfer steps, allof which involve a nonhorizontal transfer. These four can besplit into two types: from left to right either HOMO to LUMOor LUMO to HOMO. If the two halves of a coupled chro-mophore pair are not the same, there are two different matrixelements for the two types of nonhorizontal electron transfer,TD1 and TD2. Decay of the excited singlet and of the doubletriplet are considered the only ways for the excitation to leavethe system in our model. Fluorescence, internal conversion, andintersystem crossing are possible forms of singlet decay. In thecontext of solar energy conversion, charge injection into anelectrode or semiconductor nanoparticle (e.g., in a dye-sensitizedcell) is another decay pathway for the excited singlet. In thiswork we will not attempt to distinguish between these S1 decaypathways, and in Figures 4-8 we will refer to all of themtogether as “fluorescence”. All of these S1 decay pathways areincluded in the rate constants KL and KR in Figure 2. On thetime scale of these simulations, the only triplet decay pathwayis charge injection into the electrode due to (generally) longtriplet lifetimes. Triplet decay is modeled with a single rateconstant, while singlet decay is modeled with two separate decayconstants for the two types of singlets (left and right localizedsinglet). In most cases the two singlet decay rates would besimilar, but they could differ in a heterodimer or if thechromophore were bonded to the electrode differently for eachhalf of the dimer. We suggest that this simple model capturesthe essential behavior of SF, although it would certainly bepossible to add additional pathways (for example, direct couplingof the initial and final states by Coulomb and exchangeinteraction, spin flips due to spin-orbit coupling, spindipole-dipole interaction, or CT state decay).

Electronic Structure Computations. We use electronicstructure techniques to ascertain all of the parameters (energiesand matrix elements) in the ten configuration model for severalpromising coupled chromophore pairs. We choose three suchpairs and examine the effects of adding electron donating andwithdrawing groups.

To model the dynamics of TT formation, we computed theenergy of the isolated chromophores in their ground state S0,first excited singlet configurations S1, and first triplet T1, as wellas in their cation (C1) and anion (A1) ground configurations.All computations were done in the optimized ground stategeometry because we assume for convenience that the dynamicsare fast and coherent, preventing geometry relaxation (this andother effects of decoherence will be explored in future work).We continue to approximate the coupled chromophore pairenergy as a sum of two independent chromophores. Thisapproximation gives fairly accurate energies for the localizedsinglet and double triplet configurations of the dimers as longas the chromophores are weakly coupled, as all of the systemswe study are. One advantage of this simplification is that the

energy of many dimers in many different electronic configura-tions can be computed quickly. If the energy of any given pairis the sum of the energies of the two halves plus a correctionterm, one can compute energies for N2 pairs in just Ncomputations. The energy of the TT and the localized excitedsinglet configurations are generated by adding the energies oftwo independent chromophores in the appropriate electronicconfigurations:

where ETT and ES1S0are the energy of the triplet pair and

localized singlet coupled chromophore pair configurations,and ET, ES0

, and ES1are the independent chromophore triplet,

ground configuration, and first excited singlet configurationenergies. We note that this treatment (specifically eq 2) neglectsexciton coupling between the monomers, which in the J-typegeometry used here would lower the S1 energy of the dimer vsthat of the monomer.47 The change in ES1S0

caused by this excitoncoupling is expected to be small compared to the differencebetween ES1S0

and ETT or EC+A- (see below), with little effecton the calculated fission yields.

To model accurately the energy of the CT states, we mustaccount for the Coulombic stabilization from the two interactinghalves:

where EC+A- and EcC+A- are the uncorrected and corrected energy

of a CT state of a dimer, EC+, EA-, and ES0are cation, anion,

and ground energies of the monomer, and ∆ is the correctionenergy. We use constrained DFT (CDFT)48-50 to find thecorrection energy because the two charged chromophores areso close together. CDFT restricts the charge on each half of thedimer to +1 or -1 and iteratively adjusts the electron densityof each half on the basis of the field from the other half.

We run one CDFT computation for each coupled chro-mophore pair, and compare the CDFT energy (ECDFT) to theneutral dimer energy, ECCPS0

to find the correction term, ∆. Thesame correction is then applied to all variants of a given coupledchromophore pair with a given geometry. This allows a goodestimate for the Coulombic stabilization without having to runa CDFT computation on every pair of chromophores.

The electronic matrix elements for hole and electron transfer,TH and TL, of the three basic coupled chromophore pairs havebeen previously investigated with density functional theory andKoopmans’ theorem.51 The half-energy level splitting methodused to compute TH and TL can only be used directly to computeelectronic matrix elements for orbitals that are close in energy.We thus cannot use the previous model to solve for thenonhorizontal (HOMO on one chromophore to LUMO on theother chromophore) electronic matrix elements. We instead relyon the Longuet-Higgins-Roberts approximation,52-54 that

ETT ) 2ET - 2ES0(1)

ES1S0) ES1

- ES0(2)

EC+A- ) EC+ + EA- - 2ES0(3)

EC+A-c ) EC+ + EA- - 2ES0

- ∆ (4)

∆ ) (EC+ + EA- - 2ES0) - (ECDFT - ECCPS0

) (5)

14170 J. Phys. Chem. B, Vol. 114, No. 45, 2010 Greyson et al.

states that the electronic matrix element Hab is proportional tothe orbital overlap, Sab.

We are examining the overlap of orbitals that arise from theindependent chromophore Hamiltonians, but we will use themto estimate matrix elements for the coupled chromophore pairbasis. To compute the overlap, Sab, we split the geometryoptimized dimer into two chromophores by cutting the con-necting bond and hydrogen terminating each half. We thencompute the wave function of the HOMO and LUMO for eachchromophore. The (very small) overlap between the twoHOMOs and between the two LUMOs is found by integratingover space, and we use these values and the known Hab valuesfrom the half-splitting method (valid only when the two orbitalsare close in energy) to compute the proportionality constant k.The integrated overlap Sab for the nonhorizontal electron transferis then used with this k, to find Hab for nonhorizontal electrontransfer. The same k computed for a coupled chromophore pairis used to compute matrix elements for the substituted versionsof the chromophores from their overlap integrals.

All electronic structure computations were done with densityfunctional theory (DFT) using the B3LYP functional with a6-31G** basis set. Time dependent DFT (TDDFT) was usedto evaluate the excited singlets for all systems, and constrainedDFT (CDFT) was used to compute the energy of charge transferdimer configurations. All computations were done using eitherQChem 3.055 or NWChem 5.0.56,57 We used the CDFT modulewritten by Van Voorhis and co-workers for implementation inNWChem 5.0.48-50

Time Evolution of Excitation. In this paper we examinecoherent evolution of the excitation using density matrixformalism and a purely electronic Hamiltonian based on theten configuration scheme discussed previously. To estimate thepercent of the initial excitation that leaves the system throughSF as opposed to other pathways, we introduce absorbingboundary conditions.58 We investigate how the excitation canevolve in a coherent system with the Liouville-von-Neumannequation, adding in a term for dissipation.59

where H0 is the Hamiltonian for the system, HD is the dissipativematrix, and F is the density matrix. The Hamiltonian anddissipative correction matrix derived from our ten-configurationmodel are

where H0 is the Hamiltonian governing evolution within theten-configuration manifold and HD represents population leavingthe ten configuration system by fluorescence, internal conver-sion, or injection. These decay rates are expressed in units ofenergy but may be converted to lifetimes via the energy-timeuncertainty principle ∆E∆t ) p/2. The system is set up withsome initial density matrix F0, where all of the population isequally distributed among the four local S1 configurations andevolves in small finite time steps until all the population hasleft the system. At the end of the simulation we can examinethe overall fission yield, based on how much population leftthe system via triplet pair decay as well as looking at thepopulation of each determinant as a function of time. Weinterpret the percent of the excitation that has left the systemby the triplet pair decay path to represent the percent ofexcitations in a given type of dimer that would undergo SF asopposed to other decay mechanisms.

We first run several simulations using artificial, but molecu-larly relevant, configuration energies En, electronic matrixelements Tn, and decay rates Kn to understand how variationwithin the parameter space affects fission yield in this model.We subsequently use parameters computed for real dimers topredict fission yields. The combination of these sets of datasuggests both design parameters for molecular SF and systemsfor future experiments.

Results and Discussion

Electronic Structure. In this paper, we focus on one dimerof each of three different chromophores, as shown in Figure 3.1,3-Diphenylisobenzofuran (1) was chosen as a representativeof a biradicaloid system that is currently under experimental

Hab ) ⟨Ψa|H|Ψb⟩ ) kSab ) k⟨Ψa|Ψb⟩ (6)

ip(dF/dt) ) [H0,F] + [HD,F]+ (7)

H0 )

(ES1S0 0 0 0 TH TL 0 0 0 0

0 ES1S0 0 0 TL TH 0 0 0 0

0 0 ES1S0 0 0 0 TH TL 0 0

0 0 0 ES1S0 0 0 TL TH 0 0TH TL 0 0 EAC 0 0 0 0 TD2

TL TH 0 0 0 ECA 0 0 TD1 00 0 TH TL 0 0 ECA 0 0 TD1

0 0 TL TH 0 0 0 EAC TD2 00 0 0 0 0 TD1 0 TD2 ETT 00 0 0 0 TD2 0 TD1 0 0 ETT

)

HD )

(-iKL 0 0 0 0 0 0 0 0 00 -iKR 0 0 0 0 0 0 0 00 0 -iKR 0 0 0 0 0 0 00 0 0 -iKL 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 -iKTT 00 0 0 0 0 0 0 0 0 -iKTT

)

Figure 3. Structures of three promising coupled chromophore pairs.In addition to the parents (R ) H), we also examine some variantswhere R ) NH2 or NO2.

Singlet Fission in Organic Dimers J. Phys. Chem. B, Vol. 114, No. 45, 2010 14171

and theoretical examination.36,38 Pentacene (2) was chosen as arepresentative acene. For both classes of structures, there aretheoretical reasons to expect the observed favorable arrangementof energy levels, E(S1) g 2E(T1). The polyene 3 was chosenfor two reasons. Its SF would be unusually exothermic, and SFhas been suggested to explain a high triplet yield reported insome protein-bound carotenoids.21 Each of these three dimersis coupled in a similar manner, through a bond that would leadto direct conjugation if not for steric forces rotating the twochromophores nearly 90° out of plane from each other.

Each of the three primary dimers has a similar overlap integraland similar electronic matrix element owing to the similarityof the connections (Table 1). As one would expect, the electrondonating and withdrawing groups affect the overlap, and hencematrix elements of each dimer. Most notable is the strongelectron withdrawing property of the nitro groups, which leadsto nn-dimers with very low TL values. The nonhorizontal matrixelements are highly altered in the an dimers because electrontransfer from the HOMO of the aminated chromophore to theLUMO of the nitrated chromophore has very small overlap, asboth groups reduce the amplitudes of the frontier orbitals atthe coupling positions, while the overlap for a transfer fromthe HOMO of the nitrated chromophore to the LUMO of theaminated chromophores is high since now the amplitudes areenhanced. This means the two nonhorizontal electron transfermatrix elements (as well as TH and TL) are very different in thean substituted dimers as opposed to many others where mostof the electron transfer matrix elements are similar to each other.

Table 2 shows the energies of every relevant determinant inthe ten configuration model for twelve dimers (the three primarydimers and several variants carrying electron donating and

withdrawing groups). For unsubstituted species, 1 has the highestCT and TT state energies. The polyene 3 has the lowest TTenergy, with significantly exothermic SF, while in 2 SF isslightly exothermic, and it has the lowest CT energies of thethree. Derivatives of 1 and 2 carrying two amino or nitrosubstituents tend to have lower CT energies than the parent,while functionalized derivatives of 3 have higher energy CTconfigurations. The lower CT energies could result from theheteroatoms providing a good location for excess charge toreside, while it is possible that 3 is more stabilized by thefunctional groups in its local excited configuration than in thecharge transfer. Almost every functionalized dimer has TTenergy higher than the unfunctionalized version. This trend isexpected and has been discussed previously.39

Time Evolution of Model Systems: Basic Model. Tounderstand the type of dynamics that arise from this tenconfiguration coherent system, we first examine a simple casein detail. In this example, line 1 in Table 3, all ten configurationsare assumed isoenergetic, and every matrix element is equal(0.027 eV). The decay rates KL and KR for both singlet excitondeterminants and KTT the triplet pair determinant are equal andset to 2.7 × 10-4 eV, which corresponds to a lifetime of 1.2 psif singlet decay were the only pathway. The matrix elementsare close to those of several promising dimers, and the decayrates are chosen to be on the order of very fast internalconversion events, while the state energies are chosen to beisoenergetic for simplicity. For a homodimer we assume thatconfigurations 1-4 begin equally occupied.

As can be seen in Figure 4, population quickly leaves the S1

configurations, enters the CT, and then moves into the TT. Underthese conditions, the CT population reaches its first peak in lessthan 20 fs, and the TT population peaks in just under 30 fs. InFigure 4, and elsewhere, we use the term “fluorescence” tosignify all the decay channels covered by the constants KL orKR in Figure 2. Unlike an isoenergetic two-configuration fullycoherent system, where all the population would transfer to thesecond configuration before reversing direction, the TT popula-tion peaks at around 20% of the total population before all ofthe population flows back through the CT configurations to theS1 configurations. Because of the many electron transferpathways, the oscillatory nature of the populations is not simple.As expected, however, at all times, the four identical S1

configurations have the same population, as do the four identicalCT configurations, and the two identical TT configurations. Afterless than 2 ps, half the population of the system has exitedthrough one of the decay channels, and after 5 ps more than90% of the population has left. Because the decay rates areequal, the ratio of triplet pair injection to singlet decay isproportional to the ratio of the integrated populations of theTT and S1 states over time. The ratio begins low because initiallyall population is in S1, peaks as the TT population is firstbeginning to tail off (at 40 fs), and then settles on a very slowlyrising curve, yielding ∼11% fission at infinite time. Physically,this means that if a molecular system with these parameters(isoenergetic states, weak coupling and slow decay rates) wereto be made, it would exceed any known experimental result formolecular SF. Note that in the language of many experimental-ists this would be considered to be 22% yield, since two tripletsare formed for each fission event. In this paper only the formerterminology is used. To understand why the triplet yield is thispromising in this example, as well as to see if we can get higheryields (perhaps even up to 100%), we do several othersimulations with hypothetical molecular parameters below.These examples have been selected because they show some

TABLE 1: Electronic Matrix Elements for Several Bare andFunctionalized Dimersa

TH (eV) TL (eV) TD1 (eV) TD2 (eV)

1 0.068 0.068 0.066 0.0662 0.068 0.082 0.056 0.0563 0.065 0.059 0.065 0.065aa1 0.061 0.070 0.064 0.064an1 0.056 0.058 0.044 0.089nn1 0.066 0.034 0.046 0.046aa2 0.063 0.085 0.056 0.056an2 0.056 0.058 0.044 0.089nn2 0.066 0.056 0.051 0.051aa3 0.044 0.071 0.058 0.058an3 0.069 0.039 0.034 0.084nn3 0.096 0.022 0.047 0.047

a The prefixes aa, an, and nn are diamino, aminonitro, and dinitrofunctionalized variants.

TABLE 2: Energies of Electronic Configurations for SeveralCoupled Chromophore Pairs (in eV)a

S1S0 S0S1 C+A- A-C+ TT Yield (%)

1 0 0 1.005 1.005 0.303 42 0 0 0.354 0.354 -0.240 243 0 0 0.595 0.595 -0.712 1aa1 0 0 0.988 0.988 0.384 2an1 0 -0.207 -0.144 1.823 0.286 42nn1 0 0 0.898 0.898 0.397 1aa2 0 0 0.345 0.345 -0.144 42an2 0 -0.089 -0.506 1.116 -0.169 46nn2 0 0 0.353 0.353 -0.105 42aa3 0 0 0.668 0.668 -0.222 7an3 0 -0.191 -0.484 1.674 -0.257 69nn3 0 0 0.712 0.712 -0.101 21

a The yield column will be discussed at the end of this section.The origin is taken as the higher of ES0S1

and ES1S0.

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of the complex behavior of this interconnected ten configurationcoherent system. An illustrative selection of simulations issummarized in Table 3 and discussed next.

Time Evolution of Model Systems: Effect of Decay Rateson SF. One of the easiest ways to increase or decrease fissionyield within this model is to increase or decrease the ratio ofthe triplet pair and singlet decay constants, KTT, KL, and KR.Using the parameters outlined above, if KTT is doubled, thefission yield increases to 15%, while if KL and KR are insteaddoubled, the fission yield drops to 8% (Figure 5A,B, respec-tively). In both simulations the relative populations of eachconfiguration are the same, as they are controlled by the matrixelements and energy levels of the configurations. The fissionyield changes because a greater or lesser percent of thepopulation in a given configuration exits per time period whenthe decay constants are changed. If all three decay rate constantsare increased to 2.7 × 10-3 eV, the fission yield drops to 8%,because the combination of this short lifetime and the fact thatall population begins in S1 leads to a larger percent of singletdecay. If the decay rate constants are all lowered to 2.7 × 10-5

eV, however, the fission yield stays at 11%. As long as thedecay rate is significantly slower than the rate of electrontransfer, changing all of the decay constants proportionally doesnot affect fission yield (Figure 5C). Physically, this means thatlimiting nonradiative decay of singlets, singlet injection intoan electrode, and fluorescence will be important for achievinghigh fission yields in a dye sensitized solar cell. It is worth notingthat it is possible to increase KTT and have fission yield decrease.The ideal triplet pair decay rate would match the rate at whichpopulation is being fed into the TT configurations. In thecoherent isoenergetic example with all matrix elements equal,the ideal decay rate constant balances the feed-in rates to TT.A higher KTT would actually decrease fission yield, as shownin the bottom of Figure 5C. This is not true for KL and KR,which always lead to increased singlet decay when increased.This difference exists because the population begins in S1: ifwe artificially had all population begin in the TT states, theopposite would be true.

Time Evolution of Model Systems: Effect of ElectronicCoupling Matrix Elements on SF. Previous research38,39 hasexamined the strength of the electronic coupling between twochromophores in a coupled chromophore pair, since thiscoupling partly determines the rate of electron and energytransfer between the two halves. We examine how fission yieldchanges in the simple system shown in Figure 4 when the matrixelements are altered. Doubling every matrix element doublesthe rate of oscillations, but the minimum and maximumpopulations for each configuration remain unchanged, and theoverall fission yield is also unchanged (Figure S1, SupportingInformation). This result is somewhat surprising, because a firstguess would be that maximizing the electronic matrix elementwould be critical. In this example, the decay rates are far slowerthan the electron transfer rates, so the dynamic equilibriumbetween all states is established much quicker than decayregardless of the coupling in this regime. The fission yield isdetermined by the relative average populations of the S1 andTT configurations and the ratio of the singlet and triplet pairdecay rates. In both cases the average populations on S1 andTT are the same, and the only difference is the rate at whichpopulation goes back and forth. The absolute magnitude of theelectronic matrix elements is only important if there is acompeting decay pathway that occurs on the same time scale

TABLE 3: Simulations of Coherent Dynamicsa,b

figure TH TL TD1 TD2 KTT KL, KR E14 E23 E58 E67 E90 % yield

4 0.027 0.027 0.027 0.027 0.00027 0.00027 0 0 0 0 0 11.05A 0.027 0.027 0.027 0.027 0.00054 0.00027 0 0 0 0 0 15.25B 0.027 0.027 0.027 0.027 0.00027 0.00054 0 0 0 0 0 7.56A 0.027 0.054 0.027 0.027 0.00027 0.00027 0 0 0 0 0 19.06B 0.027 0.027 0.027 0.054 0.00027 0.00027 0 0 0 0 0 22.56C 0.027 0.054 0.027 0.054 0.00027 0.00027 0 0 0 0 0 28.37A 0.027 0.027 0.027 0.027 0.00027 0.00027 0 0 0.081 0.081 0 11.07B 0.027 0.027 0.027 0.027 0.00027 0.00027 0 0 0.27 0.27 0 11.08A 0.027 0.027 0.027 0.027 0.00027 0.00027 0 0 0 0 0.027 11.08B 0.027 0.027 0.027 0.027 0.00027 0.00027 0 0 0.081 0.081 0.027 7.58C 0.027 0.027 0.027 0.027 0.00027 0.00027 0 0 0.27 0.27 0.027 2.5S1A 0.054 0.054 0.054 0.054 0.00027 0.00027 0 0 0 0 0 11.0S2A 0.027 0.027 0.027 0.027 0.00027 0.00027 0.027 0 0 0 0 15.3S2B 0.027 0.027 0.027 0.027 0.00027 0.00027 0 0 0 0.027 0 16.0S2C 0.027 0.027 0.027 0.027 0.00027 0.00027 0.027 0 0 0.027 0 20.8S3A 0.027 0.027 0.027 0.027 0.0027 0 0 0 0 0 0 25.0S3B 0.027 0.027 0.027 0.027 0.0027 0 N/A 0 0 0 0 50.0S3C 0.027 0.027 0.027 0.027 0.0027 0 N/A, 0 N/A N/A, 0 N/A N/A, 0 100.0

a All couplings, decay rates, and energies are listed in eV. b Each row corresponds to a different simulation, and is identified by the numberof a figure that provides more details. The electronic coupling matrix elements are listed first, then the decay rate constants, followed by theenergies of the configuration (E14 stands for configuration 1 and 4), and finally the fission yield is listed. For additional detail on thesimulations, see text.

Figure 4. Coherent dynamics simulation of a basic isoenergetic system.Time dependence of the sum of all populations in S1 (blue solid), CT(red solid), or TT (green solid) configurations, the percentage of thepopulation that has left via fission (purple dash-dot) or fluorescence(orange dash) routes, and the percentage of population that has left thesystem through fission vs fluorescence (black solid). The inset showsthe first 0.1 ps.

Singlet Fission in Organic Dimers J. Phys. Chem. B, Vol. 114, No. 45, 2010 14173

as electron transfer. If decay were very quick (<100 fs),significant population could decay from S1 before any populationreached TT. In this case larger electronic matrix elements wouldbe important.

When only some of the electronic matrix elements are altered,however, the fission yield changes even when the decay rate isleft the same (KL ) KR ) KTT ) 2.7 × 10-4 eV). If either theTH or the TL matrix element is doubled, but the other elementsare unchanged from the initial simulation, the fission yieldincreases to ∼19%. (Figure 6A). This increase in fission yieldoccurs because the destructive interference has been reduced,and some of the pathways (TH vs TL) from S1 to CT areoscillating on a different time scale. The impact of destructiveinterference is discussed further in the Supporting Information.In a similar vein, if all matrix elements are the same, exceptfor either TD1 or TD2, then the yield jumps to 23% (Figure 6B).Additionally, the fission yield increases as the ratio of TH to TL

increases to 2, but if the ratio of the matrix elements increasesfurther, the fission yield drops again, to 15% when TH ) 3TL.If both one of TH or TL and one of TD1 or TD2 are doubled whilethe other two remain the same, the fission yield jumps to 28%,as shown in Figure 6C (and if KL ) KR in this scenario, 100%fission yield is realized). In contrast to these beneficial modifica-tions, if TH and TL are doubled but TD1 or TD2 is left unchanged,the fission yield drops to 5%. This is easy to understand, as theoscillations from S1 to CT have sped up owing to the increasedmatrix elements, but the CT to TT transitions have not. Thesetwo transitions get out of phase, resulting in more population

in S1 and CT and less in TT compared to the case when allmatrix elements are equal.

In terms of physical meaning, these examples provide severalpossible strategies for constructing an ideal dimer. Fission yieldis increased when TH and TL are not equal. This will be thecase when the dimer has more coupling (overlap) between thetwo chromophores in either the HOMO or LUMO. Previouslyexamined molecules that displayed this tendency were donor-acceptor type structures. Fission yield is also increased whenthe two nonhorizontal electron transfer elements are not equal.To realize this, the dimer should be designed such that theHOMOmol1-LUMOmol2 overlap is better than the HOMOmol2-LUMOmol1 overlap, or vice versa. This could be done either byconnecting two identical chromophores to each other in differentlocations or by binding two different chromophores together tomake a heterodimer.

Effects of Nondegenerate State Energies on SF. Until nowwe have discussed highly artificial systems where all configura-tions are isoenergetic. This has let us study some of the behaviorof our ten configuration coherent system but is not realistic fora real molecular system. We now examine the effects ofchanging the energy levels of various configurations usingparameters similar to those in previous simulations, with allmatrix elements set to 0.027 eV, and all decay rate constantsequal to 2.7 × 10-4 eV. When the energies of both of the CTconfigurations are increased above that of the S1 and TTconfigurations, there is no change in fission yield for small CTenergies (Figure 7A). As the energy of the CT configurations

Figure 5. Coherent dynamics simulations showing the effects of doubling the TT decay (A) or S1 decay (B) vs Figure 4. (C) shows the effect ofa range of S1 and TT decays on the fission yield.

Figure 6. Coherent dynamics simulations of the effect of asymmetric electronic coupling matrix elements: (A) only TL doubled; (B) only TD1

doubled; (C) TL and TD1 doubled.

14174 J. Phys. Chem. B, Vol. 114, No. 45, 2010 Greyson et al.

moves further away from the S1 and TT configurations, thereare two noticeable differences in the population of variousconfigurations over time. The CT configurations have a smallerpopulation as the energy gap increases, and the major periodof oscillation for all three types of configurations increases(Figure 7B). This is similar to coherent electron transfer betweentwo configurations of different energies: as the energy gapincreases, the period of oscillation increases, and the amountof population that is transferred decreases. Importantly, however,the relative populations of S1 and TT are unaffected by thisenergy barrier. Given isoenergetic S1 and TT configurations,the CT barrier height only affects the fission yield if the barrierheight and singlet decay rate are such that all population leavesthe system (from S1) before one full period of S1 to TToscillations. At a barrier of 0.27 eV, the singlet would have todecay completely (by fluorescence or injection, for example)on the order of 90 fs to have the fission yield affected by thecharge transfer energy barrier. If the barrier were 1.5 eV, thefission yield would decrease if the singlet exciton decayed inless than 750 fs. If both CT configurations are an equal energybelow the S1 and TT configurations instead of above them, theexact same behavior in population over time and overall fissionyield is seen. In the regime of fast coherent electron transferthe energy difference between configurations (and matrixelements) determine the rate and magnitude of electron transfer,and the direction of energy change (uphill vs downhill) isirrelevant. This is typical of purely coherent superexchange

phenomena and it should be noted that this is very differentfrom how electron transfer rates depend on the energy in aMarcus-type rate picture.60-62

If S1 and CT are isoenergetic but TT is shifted in energy, thefission yield decreases, regardless of whether the product TT ishigher or lower in energy. From 11% fission yield in theisoenergetic case, the fission yields drops to 5% with a 0.081eV offset, and <1% with a 0.27 eV offset. If CT and TT areisoenergetic, and S1 is shifted instead, there is an identical dropin fission yield. If we combine an elevated CT configurationenergy with a ∆G between the S1 and TT configurations, thefission yield is worse than if there were the same ∆G but noCT barrier. In this case, the height of the CT barrier does affectfission yield. For a ∆G of 0.027 eV, the fission yield is 11%with no barrier, 8% with a 0.081 eV barrier, and 2.5% with a0.27 eV barrier (Figure 8A-C, respectively). The main reasonthe barrier affects fission yield in this case is because the twoelectron transfer steps oscillate with different periods. WhileCT configurations at 0.081 eV and TT configurations at 0.027eV lead to only 8% fission yield, if the CT barrier is put at-0.081 eV, the yield is 11%. When the barrier is positive, theenergy difference between S1 and CT is twice the gap betweenCT and TT, so the oscillations are very out of phase, whereasa negative barrier puts the oscillations of the S1 to CT electrontransfer step fairly close to the CT to TT electron transfer step,providing a higher fission yield.

Figure 7. Coherent dynamics simulations of the effect of low (A) and high (B) CT barrier heights.

Figure 8. Coherent dynamics simulations of the effect of nondegenerate S1 and TT energies with zero (A), low (B), or high (C) CT barrier heights.

Singlet Fission in Organic Dimers J. Phys. Chem. B, Vol. 114, No. 45, 2010 14175

In a heterodimer, it would be possible for the left and rightlocalized S1 and charge transfer configurations to be different inenergy. If one of two sets of localized S1 (configurations 1,4 vs2,3) or CT (configurations 5,8 vs 6,7) configurations is not equalin energy to the other, the fission yield is typically increased. Ifeither the left (or right) localized S1 (or CT) configurations areincreased in energy by 0.027 eV, and all other parameters areunchanged, the fission yield increases from 11% to 16% (FigureS2A,B, Supporting Information). If both one set of localized singletconfigurations and one set of localized CT configurations are raised0.027 eV, the fission yield increases to 21% (Figure S2C,Supporting Information). Changing the initial population of thesystem will change the fission yield in any simulation where thetwo localized singlets are not isoenergetic. If all of the populationbegins in the higher energy set of S1 configurations in theseexamples, the yields are 11% and 15%. If all of the populationbegins in the S1 configurations with lower energy (and isoenergeticwith TT), the fission yields are 20 and 27%. In all of these casesthe enhanced fission yield comes from symmetry breaking. In theunphysical case where KL ) KR ) 0, the fission yield is 50% ifeither the S1 or CT configurations have broken symmetry, and100% if both sets of configurations have broken symmetry (seediscussion of interference effects in the Supporting Information).

From a molecular design point of view, these simulationssuggest that heterocoupled chromophore pairs may be attractivetargets because the two sets of localized S1 and CT configura-tions will have different energies (although heteroCCP’s willalso have less favorable S1-TT gaps). It is worth noting thatunlike the case of the electronic matrix elements, even a verysmall breaking of the symmetry leads to a large increase infission yield. Because of this, small differences in chromophores,such as electron donating or withdrawing groups, might suffice.

Lessons Learned. One of the major limitations for fission yieldis the destructive interference that occurs when the system has highsymmetry. This symmetry (and interference) is broken when leftand right localized S1 configurations and CT configurations havedifferent energies, as is expected to be the case for heterodimers.The symmetry can also be broken by having the matrix elementsTH and TL unequal, and TD1 and TD2 unequal. Additionally, thissymmetry could be broken in a real system by dephasing or by afluctuating environment. A second major factor in the fission yieldis the relative decay rates of the TT and S1. Because either half ofa TT configuration can inject into the electrode first, while onlythe excited S1 half of S1S0 can inject, it might be beneficial to linkone-half of a dimer to the electrode more strongly than the other.If it is a heterodimer, by linking the higher energy localized S1 tothe electrode one might be able to retard singlet exciton injectioninto the electrode without adversely affecting triplet-triplet injec-tion. The third major conclusion we can draw from these simula-tions is that coherent dynamics with no dephasing and zerotemperature suggest that having the ∆G of SF as close to zero aspossible will help fission yield (note that this is a stricter conditionthan the commonly noted ∆G < 0 requirement). This conclusionagrees with the exponential gap law derived by Singh fromconsideration of the Coulomb and exchange interaction.41

Time Evolution of Real Molecules. Table 2 shows the fissionyield for each of the twelve coupled chromophore pairs found usingcoherent dynamics within this ten configuration scheme. All ofthese simulations were run with KL ) KR ) 2.7 × 10-7 eV (1.2ns lifetime), while KTT ) 2.7 × 10-4 eV (1.2 ps lifetime). Theenergy parameters in Table 3 were taken from Tables 1 and 2.These parameters are chosen to represent fluorescent decay of thelocalized singlets, and a quick triplet pair injection into an electrode.All three homodimers of 1 display very poor SF yield. They all

have very high energy charge transfer configurations, and becauseof this very little population enters the CT configurations. The an-1chromophore has one CT configuration very close in energy toone of the S1 configurations and, as a result, has a dramaticallyhigher SF yield.

The most distinct trend in this data is that the dimers with TTenergies closest to their lowest S1 energies have the highest SFyield. In order of increasing absolute value of free energy of fission,the chromophores are ranked an3, an2, nn3, nn2, aa2, aa3, 2, 1,aa1, nn1, an1, and 3. Only three of the dimers in this list are notin the place we would expect if we were to approximate that theonly factor leading to high fission yield was a small absolute valueof free energy of fission (energy of TT-S0S1). aa3 and nn3 wouldboth be expected to have a greater SF yield based solely on theirfree energies of fission; however, their CT energy configurationsare roughly twice as far away from S1 and TT configurations asany of the more efficient dimers. an1 would be expected to beless efficient at SF based solely on the free energy of fission, butwe note that it has a CT level far closer to its S1 energy level thanany other chromophore. This means more population transfers fromthe S1 to the CT configuration, which limits singlet decay, andthus produces greater population in TT.

Overall, having a free energy of fission close to zero was themost important predictor of high fission yield, and having CTclose in energy to S1 and TT was the second most importantfactor. The electronic coupling matrix elements do not seem tobe as important within the range that they cover in thesemolecules. This is what one would expect on the basis of themodel simulations. We also note that the particular values ofS1 and TT decay chosen are unimportant when the relative yieldsbetween two coupled chromophore pairs with similar decay ratesbut different energetics are compared. As noted earlier, thefission yield is unchanged if the two decay rates are scaled bythe same amount, as long as they are slower than the initialequilibration process between S1, CT, and TT (Figure 5C).

Summary and Conclusion

For SF to be efficient within the framework of coherent, purelyelectronic dynamics, it is critical for the S1 and TT states to be asclose to isoenergetic as possible. If TT is not close in energy tothe initial S1 energy, the time average population in TT is verylow, leading to low fission efficiency. In addition to this constrainton the free energy of the reaction, it is essential for CT to lierelatively close to the S1 and TT energies for the population to beable to oscillate back and forth from S1 to TT on a time scalecomparable to or faster than excitation decay (as discussed earlier,we have neglected direct coupling of the initial and final states bydirect Coulomb and exchange interaction). The magnitude of thevarious electron transfer matrix elements is not as important asthe configuration energies, and thus dimer design should be donewith energy in mind first. In most cases this will mean a relativelyweak coupling is preferable, to prevent fission from being highlyendothermic, but some chromophores, such as twisted polyenes,might perform better with stronger coupling because of low tripletenergies. The energy of the CT intermediates can be controlledboth by the choice of chromophore, where using biased chro-mophores provides a reduction in CT energy, and by controllingthe Coulombic stabilization of a dimer CT configuration bymoderating the distance between the charged chromophores. Wefind that dimers that combine these strategies to place as manyenergy levels as close together as possible are the ones that showthe highest SF yields in the regime of coherent electron dynamics.

Several changes would be expected if dephasing were includedin our model. Perhaps most importantly, the parasitic interference

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that dramatically lowers singlet fission in some instances woulddisappear. We are currently investigating singlet fission using the(incoherent) Marcus electron transfer model and hope to discussthe similarities and differences of these two approximations.

Acknowledgment. We are grateful to the chemistry divisionof the ONR (N00014-05-1-0021) and to the DOE (1542544/XAT-5-33636-01, DE-FG36-08GO18017) for support of this work, andto Professor Abraham Nitzan and Dr. Sina Yeganeh for usefulsuggestions.

Supporting Information Available: Figures S1-S3: simula-tions of the effect of doubling the electronic coupling matrixelements; simulations of hetero-CCPs with nondegenerate energylevels; exploration of interference effects. Discussion of the effectof destructive interference on fission yields. This material isavailable free of charge via the Internet at http://pubs.acs.org.

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